This invention relates to a mixing and separating method for a plurality signals, more particularly, to achieve control circuit for signal processing.
The electronic communication has grown more and more prosperous and the problem of available of communication channel becomes even serious. The bandwidth of a specific communication medium is limited, therefore how to most exploit the avail bandwidth is essential. It is the object of the invention to provide a method for transmitting a plurality of signals in a pair of transmission line or a single channel.
The present invention is based on the principle of unique solution condition for a set of N linearly equations, i.e., linearly independence. Based on this principle, each of the m signal Si(t) within period [T0, T1] is sampled for n samples Si(tj), j=1,2 . . . n, wherein txcex5[T0, T1], T0, T1 xcex5R, t is time variable. Each sample is multiplied by a coefficient function iaj(t) which is a linear independent set (i=1,2 . . . m, j=1,2 . . . n), thus obtaining m transformed signals for Si(t):             S      i      0        ⁢          (      t      )        =            ∑              j        =        1            n        ⁢          xe2x80x83        ⁢          [                                    a            j                                      i                                  ⁢                      (            t            )                          ⁢                              S            i                    ⁢                      (                          t              j                        )                              ]      
summing above m transformed signals obtains the mixed transformed signals:       SM    ⁢          (      t      )        =            ∑              i        =        1            m        ⁢          xe2x80x83        ⁢          [                        S          i          0                ⁢                  (          t          )                    ]      
The mixed transformed signals have mxc3x97n variables Si(tj) with coefficient iaj(t). If party A transmits SM(t) during time [T0, T1] to party B, party B will obtain message of mxc3x97n Si(tj), (i=1,2 . . . m, j=1,2 . . . n), wherein the bandwidth depends on the max bandwidth of iaj(t). More particularly, party A can m messages Si(t) (i=1,2 . . . m) to party B during time [T0, T1], if the samples (unknowns) Si(t1), Si(t2), Si(t3) . . . Si(tn), ((i=1,2 . . . m) are sufficient to represent Si(t) (i=1,2 . . . m) during time [T0, T1].
The party B can resolve Si(ti) upon receiving SM(t) if mxc3x97nxe2x88x921 differential means are provided to obtain differential signals SMxe2x80x2(t), SMxe2x80x3(t) . . . SMmxc3x97nxe2x88x921(t):                                                         ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                      t                      )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (              t              )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                      xe2x80x2                                              i                                                                ⁡                                          (                      t                      )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    SM              xe2x80x2                        ⁡                          (              t              )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                              (                                                                              m                            xc3x97                            n                                                    -                          1                                                )                                                                    i                                                                ⁡                                          (                      t                      )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    SM                                                m                  xc3x97                  n                                -                1                                      ⁡                          (              t              )                                                          (        1        )            
Eq( 1) is an mxc3x97n equation set, wherein functions iaj(t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent. Therefore, Si(ti) in Eq(1) has unique solution because the Wronskin (determinant)of functions iaj(t) is not equal to zero.
Therefore, the Si(ti) can be calculated by choosing a specific time t0 within [T0, T1] and obtain iaj(u)(t0) and SM(u)(t0). Moreover, each Si(t) can be calculated (i=1,2 . . . m,j=1,2 . . . n, u=0, 1,2 . . . mxc3x97nxe2x88x921).
The above solving procedure requires mxc3x97nxe2x88x921 differential means to solve Si(t), the hardware structure is bulky. However, party B can also take mxc3x97n samples after receiving SM(t):                                                         ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                1                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                2                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                                                  m                          xc3x97                          n                                                                    )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                                  m                  xc3x97                  n                                            )                                                          (        2        )            
wherein t1, t2 . . . tmxc3x97n are all within [T0, T1] and tuxe2x89xa0tv if uxe2x89xa0v, (u, v=0,1,2 . . . mxc3x97n). Si(t) has unique solution because iaj(t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent in [T0, T1].
In above scheme, party requires to take mxc3x97n samples within [T0, T1] even thought the differential means can be saved. The sample frequency will increase when the number of signal (m) increases. Therefore, the sampling rate of the A/D should be considered to determine the number of signal m.
To increase m and keep hardware compact, a compromise is to use mxe2x88x921 differential means to get m differential signals (including original SM(t)), and to take n samples for each signal within [T0, T1] thus obtaining following equation set:                                                         ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                1                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                2                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                n                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                      xe2x80x2                                              i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    SM              xe2x80x2                        ⁡                          (                              t                1                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                      xe2x80x2                                              i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    SM              xe2x80x2                        ⁡                          (                              t                2                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                      xe2x80x2                                              i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    SM              xe2x80x2                        ⁡                          (                              t                n                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                              (                                                  m                          -                          1                                                )                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    SM                              (                                  m                  -                  1                                )                                      ⁡                          (                              t                n                            )                                                          (        3        )            
iaj(t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent. Therefore, Si(ti) in Eq(3) has unique solution because the Wronskin (determinant) of functions iaj(t) is not equal to zero.
Party B has a plurality of ways to create mxc3x97n linear independent equation set form SM(t) as will be described below.                                                         ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                1                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                2                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                n                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      1                    D                                    ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    1              D                        ⁢                          SM              ⁡                              (                                  t                  1                                )                                                    ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      1                    D                                    ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    1              D                        ⁢                          SM              ⁡                              (                                  t                  2                                )                                                    ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      1                    D                                    ⁢                                                            a                      j                      xe2x80x2                                              i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    1              D                        ⁢                          SM              ⁡                              (                                  t                  n                                )                                                    ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      1                                          D                                              m                        -                        1                                                                              ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    1                              D                                  m                  -                  1                                                      ⁢                          SM              ⁡                              (                                  t                  n                                )                                                                        (        4        )            
wherein       1          D      u        ⁢            a      j                    i                ⁢          (              t        v            )      
xe2x80x83and       1          D      u        ⁢      SM    ⁢          (              t        v            )      
xe2x80x83is uth integration of iaj(t) and SM(t) from 0 to tv., u=1,2 . . . mxe2x88x921, v=1,2 . . . n.
Another alternative is:                                                         ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                1                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                2                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                n                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                  Δ                  ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      Δ            ⁢                          xe2x80x83                        ⁢                          SM              ⁡                              (                                  t                  1                                )                                                    ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                  Δ                  ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      Δ            ⁢                          xe2x80x83                        ⁢                          SM              ⁡                              (                                  t                  2                                )                                                    ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                  Δ                  ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      Δ            ⁢                          xe2x80x83                        ⁢                          SM              ⁡                              (                                  t                  n                                )                                                    ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      Δ                                          m                      -                      1                                                        ⁢                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    Δ                              m                -                1                                      ⁢                          SM              ⁡                              (                                  t                  n                                )                                                                        (        5        )            
wherein xcex94uiaj(tv) and xcex94uSM(tv) is uth differential of iaj(t) and SM(t) at tv, u=1,2 . . . mxe2x88x921, v=1,2 . . . n.
Still another alternative is:                                                         ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        1                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                1                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        2                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                2                            )                                      ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            a                      j                                                                    i                                                                ⁡                                          (                                              t                        n                                            )                                                        ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      SM            ⁡                          (                              t                n                            )                                      ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      ∇                                                                  a                        j                                                                          i                                                                      ⁡                                              (                                                  t                          1                                                )                                                                              ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      ∇                          SM              ⁡                              (                                  t                  1                                )                                                    ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      ∇                                                                  a                        j                                                                          i                                                                      ⁡                                              (                                                  t                          2                                                )                                                                              ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      ∇                          SM              ⁡                              (                                  t                  2                                )                                                    ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                      ∇                                                                  a                        j                                                                          i                                                                      ⁡                                              (                                                  t                          n                                                )                                                                              ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                      ∇                          SM              ⁡                              (                                  t                  n                                )                                                    ⁢                  
                ⁢        ⋮        ⁢                  
                ⁢                                            ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  j                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                              [                                                                            ∇                                              m                        -                        1                                                              ⁢                                                                  a                        j                                                                          i                                                                      ⁡                                              (                                                  t                          n                                                )                                                                              ⁢                                                            S                      i                                        ⁡                                          (                                              t                        j                                            )                                                                      ]                                              =                                    ∇                              m                -                1                                      ⁢                          SM              ⁡                              (                                  t                  n                                )                                                                        (        6        )            
wherein ∇uiaj(tv) and ∇uSM(tv) is uth summation of iaj(t) and SM(t) from 0 to tv. u=1,2 . . . mxe2x88x921, v=1,2 . . . n.
The determinant in each matrix of Eqs(4)-(6) is not zero because iaj(t) (i=1,2 . . . m, j=1,2 . . . n) are linear independent in [T0, T1]. Therefore, S1(t) has unique solution for Eqs (4)-(6).
Moreover, party B can mix the operations of differential, integration, difference, summation and sampling to create mxc3x97n linear independent equation set. For example, taking differential, integration, difference, summation for number of r1, r2,r3 and r4, and taking sample number of h, such that (r1+r2+r3+r4)h=mxc3x97n, party B can create mxc3x97n linear independent equation set for solving Si(tj). However, the other methods are not described here for clarity.
A particular choice of iaj(t) is described below, wherein iaj(t) thus selected are orthonormal for t within period [T0, T1]             ∫              T        0                    T        1              ⁢                            a          l                                k                            ⁡                  (          x          )                    ⁢                        a          j                                i                            ⁡                  (          x          )                    ⁢              xe2x80x83            ⁢              ⅆ        x              =      {                                        1            ;                          xe2x80x83                        ⁢                                          when                ⁢                                  xe2x80x83                                ⁢                k                            =                                                i                  ⁢                                      xe2x80x83                                    ⁢                  and                  ⁢                                      xe2x80x83                                    ⁢                  l                                =                j                                                                                      0            ;                          xe2x80x83                        ⁢                                          when                ⁢                                  xe2x80x83                                ⁢                k                            ≠                              i                ⁢                                  xe2x80x83                                ⁢                or                ⁢                                  xe2x80x83                                ⁢                l                            ≠              j                                          
wherein i,k=1,2 . . . m, 1,j=1,2 . . . n.
in this situation             ∫              T        0                    T        1              ⁢                  "AutoLeftMatch"                  SM          ⁢                      (            x            )                          "AutoRightMatch"            ⁢                        a          l                                k                            ⁢                  (          x          )                    ⁢              xe2x80x83            ⁢              ⅆ        x              =                    ∫                  T          0                          T          1                    ⁢                        [                                    ∑                              i                =                1                            m                        ⁢                          xe2x80x83                        ⁢                                          ∑                                  J                  =                  1                                n                            ⁢                              xe2x80x83                            ⁢                                                                    a                    j                                                              i                                                          ⁢                                      (                    x                    )                                                  ⁢                                                      s                    i                                    ⁢                                      (                                          t                      j                                        )                                                                                ]                ⁢                              a            l                                      k                                  ⁢                      (            x            )                          ⁢                  xe2x80x83                ⁢                  ⅆ          x                      =                  S        k            ⁢              (                  t          l                )            
Therefore, when the transformed and mixed signal SM(t) is sent to the receiver during time period [T0, T1] the receiver party multiplies the received signal with kal(t) (k=1,2 . . . m, 1=1,2 . . . n) and integrates the result between time period [T0, T1] to obtain Sk(tj) (the Ith sample for the kth signal). This indicates that each sample value for each signal Si(tj) can be calculated without the step of solving the linear algebraic equation set.
Below describes the way to orthonormalize the function group {iaj(t) (i=1,2 . . . m, j=1,2 . . . n)} within time period [T0, T1].
First, mxc3x97n functions G1(t), G2 (t) . . . Gmxc3x97n(t) linearly independent within time period [T0, T1] are selected and let                               h          ⁡                      (                          r              ,              s                        )                          =                              ∫                          T              0                                      T              1                                ⁢                                                    G                r                            ⁡                              (                x                )                                      ⁢                                          G                s                            ⁡                              (                x                )                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              x                                          ;              xe2x80x83            ⁢      r        ,          s      =      1        ,          2      ⁢              xe2x80x83            ⁢      …      ⁢              xe2x80x83            ⁢      m      xc3x97      n                  A      0        =    1              A      v        =                            "LeftBracketingBar"                                                                      h                  ⁡                                      (                                          1                      ,                      1                                        )                                                                                                h                  ⁡                                      (                                          1                      ,                      2                                        )                                                                                                h                  ⁡                                      (                                          1                      ,                      3                                        )                                                                              …                                                              h                  ⁡                                      (                                          1                      ,                      v                                        )                                                                                                                        h                  ⁡                                      (                                          2                      ,                      1                                        )                                                                                                h                  ⁡                                      (                                          2                      ,                      2                                        )                                                                                                h                  ⁡                                      (                                          2                      ,                      3                                        )                                                                              …                                                              h                  ⁡                                      (                                          2                      ,                      v                                        )                                                                                                                        h                  ⁡                                      (                                          3                      ,                      1                                        )                                                                                                h                  ⁡                                      (                                          3                      ,                      2                                        )                                                                                                h                  ⁡                                      (                                          3                      ,                      3                                        )                                                                              …                                                              h                  ⁡                                      (                                          3                      ,                      v                                        )                                                                                                      …                                            …                                            …                                            …                                            …                                                                                      h                  ⁡                                      (                                          v                      ,                      1                                        )                                                                                                h                  ⁡                                      (                                          v                      ,                      2                                        )                                                                                                h                  ⁡                                      (                                          v                      ,                      3                                        )                                                                              …                                                              h                  ⁡                                      (                                          v                      ,                      v                                        )                                                                                "RightBracketingBar"                ⁢                  xe2x80x83                ⁢        v            =              1.2        ⁢                  xe2x80x83                ⁢        …        ⁢                  xe2x80x83                ⁢        m        xc3x97        n            
then establishing the function             A      v        =          "LeftBracketingBar"                                                  h              ⁡                              (                                  1                  ,                  1                                )                                                                        h              ⁡                              (                                  1                  ,                  2                                )                                                                        h              ⁡                              (                                  1                  ,                  3                                )                                                          …                                              h              ⁡                              (                                  1                  ,                  u                                )                                                                                        h              ⁡                              (                                  2                  ,                  1                                )                                                                        h              ⁡                              (                                  2                  ,                  2                                )                                                                        h              ⁡                              (                                  2                  ,                  3                                )                                                          …                                              h              ⁡                              (                                  2                  ,                  u                                )                                                                          …                                …                                …                                …                                …                                                              h              ⁡                              (                                                      u                    -                    1                                    ,                  1                                )                                                                        h              ⁡                              (                                                      u                    -                    1                                    ,                  2                                )                                                                        h              ⁡                              (                                  u                  -                  1.3                                )                                                          …                                              h              ⁡                              (                                                      u                    -                    1                                    ,                  u                                )                                                                                                        G                1                            ⁡                              (                t                )                                                                                        G                2                            ⁡                              (                t                )                                                                                        G                3                            ⁡                              (                t                )                                                          …                                                                                G                  u                                ⁡                                  (                  t                  )                                            )                                          "RightBracketingBar"            u    =          1.2      ⁢              xe2x80x83            ⁢      …      ⁢              xe2x80x83            ⁢      m      xc3x97      n      
then the function                               l          u                ⁢                  (          t          )                    =                        1                                                    A                                  u                  -                  1                                            ⁢                              A                u                                                    ⁢                              P            u                    ⁢                      (            t            )                                ;          xe2x80x83        ⁢          u      =      1        ,      2    ⁢          xe2x80x83        ⁢    …    ⁢          xe2x80x83        ⁢    m    xc3x97    n  
is orthonormal within time period [T0, T1].
Moreover, assume iaj(t)=Q(Ixe2x88x921)n+j(t) (i=1,2 . . . m, j=1,2 . . . n), that is
1a2(t)=Q1(t)
1a2(t)=Q2(t)
xe2x80xa2
xe2x80xa2
xe2x80xa2
1an(t)=Qn(t)
2a1(t)=Q1+n(t)
2a2(t)=Q2+n(t)
xe2x80xa2
xe2x80xa2
xe2x80xa2
2an(t)=Q2n(t)
3a1(t)=Q1+2n(t)
3a2(t)=Q2+2n(t)
xe2x80xa2
xe2x80xa2
xe2x80xa2
3a2(t)=Q3n(t)
xe2x80xa2
xe2x80xa2
xe2x80xa2
man(t)=Qmxc3x97n(t)
Apparently, function group {iaj(t) (i=1,2 . . . m, j=1,2 . . . n)} are orthonormal within time period [T0, T1]. In above procedure for receiver party to restore Sk(t1), the step of solving linear algebraic equation is eliminated. However, mxc3x97n integrals are required, this will make the hardware complicated. The present invention provide following approach.
By choosing suitable function group {Gu(t)|u=1,2 . . . mxc3x97n}, ka1(t) can be expressed into Power series as following:                                           a            l                                      k                                  ⁡                      (            t            )                          =                  xe2x80x83                ⁢                                            a              l                                            k                                        ⁡                          (              b              )                                +                      xe2x80x83                    ⁢                                                    a                l                xe2x80x2                                  k                                              ⁡                              (                b                )                                      ⁢                          xe2x80x83                        ⁢                          t              ⁡                              (                                  t                  -                  b                                )                                              +                      xe2x80x83                    ⁢                                    1                              2                !                                      ⁢                          xe2x80x83                        ⁢                                          a                l                xe2x80x3                                  k                                              ⁡                              (                b                )                                      ⁢                          xe2x80x83                        ⁢                                          (                                  t                  -                  b                                )                            2                                +                                                  xe2x80x83                ⁢                                            1                              3                !                                      ⁢                                          a                l                xe2x80x2xe2x80x2xe2x80x2                                  k                                              ⁡                              (                b                )                                      ⁢                                          (                                  t                  -                  b                                )                            3                        ⁢                          xe2x80x83                        ⁢            …                    ⁢                      xe2x80x83                    ;                      xe2x80x83                    ⁢                      b            ∈                          xe2x80x83                        ⁢            R                              
Because the high frequency components of kal(t) are limited, the first several terms are sufficiently to represent kal(t). Assuming that first M terms are considered here, kal(t) can be expressed as:                     a        l                          k                      ⁢              (        t        )              ≅                  ∑                  q          =          0                          M          -          1                    ⁢              xe2x80x83            ⁢                        [                                    1                              q                !                                      ⁢                                          a                l                                  (                  q                  )                                                  k                                              ⁢                              (                b                )                                      ⁢                                          (                                  t                  -                  b                                )                            q                                ]                ⁢                  xe2x80x83                ⁢        b              ∈  R
then                                                                                           S                  k                                ⁡                                  (                                      t                    l                                    )                                            =                              xe2x80x83                            ⁢                                                ∫                                      T                    0                                                        T                    1                                                  ⁢                                                                            SM                      ⁡                                              (                        x                        )                                                              k                                    ⁢                                                            a                      l                                        ⁡                                          (                      x                      )                                                        ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    x                                                                                                                          =                              xe2x80x83                            ⁢                                                ∑                                      q                    =                    0                                                        M                    -                    1                                                  ⁢                                  xe2x80x83                                ⁢                                  [                                                            1                                              q                        !                                                              ⁢                                                                  a                        l                                                  (                          q                          )                                                                          k                                                                      ⁡                                              (                        b                        )                                                              ⁢                                                                  ∫                                                  T                          0                                                                          T                          1                                                                    ⁢                                                                        SM                          ⁡                                                      (                            x                            )                                                                          ⁢                                                                              (                                                          x                              -                              b                                                        )                                                    q                                                ⁢                                                  xe2x80x83                                                ⁢                                                  ⅆ                          x                                                                                                      ]                                                                                        (        7        )            
The receiver party only need to store mxc3x97nxc3x97M data             1              q        !              ⁢                  a        l                  (          q          )                          k                      ⁡              (        b        )              ⁢          xe2x80x83        ⁢          (                        k          =          1                ,                  2          ⁢                      xe2x80x83                    ⁢          …          ⁢                      xe2x80x83                    ⁢          m                ,                  l          =          1                ,                  2          ⁢                      xe2x80x83                    ⁢          …          ⁢                      xe2x80x83                    ⁢          n                ,                  xe2x80x83                ⁢                  q          =          0                ,                              1            ⁢                          xe2x80x83                        ⁢            …            ⁢                          xe2x80x83                        ⁢            M                    ;                      b            ⁢                          xe2x80x83                        ∈            R                              )        ,
then calculate M integrals       ∫          T      0              T      1        ⁢            SM      ⁢              (        x        )              ⁢                  (                  x          -          b                )            q        ⁢          xe2x80x83        ⁢          ⅆ      x      
upon receiving the SM(t). The mxc3x97n samples Sk(t1) are restored. In other word, receiver party only requires to prepare M integrators other than mxc3x97n.
Moreover, above approach is also applicable to the data compress technology. The mxc3x97n samples Sk(t1) can be approximately represented by M data:                     ∫                  T          0                          T          1                    ⁢                        SM          ⁢                      (            x            )                          ⁢                              (                          x              -              b                        )                    q                ⁢                  xe2x80x83                ⁢                  ⅆ          x                      ;          xe2x80x83        ⁢          q      =      0        ,  1  ,                    2        ⁢                  xe2x80x83                ⁢        …        ⁢                  xe2x80x83                ⁢        M            -      1        ;          xe2x80x83        ⁢          b      ∈      R      
In other word, the function group {iaj(t) (i=1,2 . . . m, j=1,2 . . . n)} are orthonormalized within time period [T0, T1] by above procedure. The M data;                     ∫                  T          0                          T          1                    ⁢                        SM          ⁢                      (            x            )                          ⁢                              (                          x              -              b                        )                    q                ⁢                  xe2x80x83                ⁢                  ⅆ          x                      ;          xe2x80x83        ⁢          q      =      0        ,  1  ,                    2        ⁢                  xe2x80x83                ⁢        …        ⁢                  xe2x80x83                ⁢        M            -      1        ;          xe2x80x83        ⁢          b      ∈      R      
are calculated by using known mxc3x97n samples Sk(t1). Afterward, Eq(7) is employed to restore mxc3x97n samples Sk(t1), wherein       ∫          T      0              T      1        ⁢            SM      ⁢              (        x        )              ⁢                  (                  x          -          b                )            q        ⁢          xe2x80x83        ⁢          ⅆ      x      
are the data after compression and       1          q      !        ⁢            a      l              (        q        )                    k                ⁢          (      b      )      
refers to the restoring parameter. It is apparent that the invention is applicable both to transmission of mass data or the compression technology of signal
From above description, party A sends message to party B segment by segment with time duration [T0, T1]. However, party A need to send a synchronous signal before sending signal containing message.
Therefore, the duration [T0, T1] is divided into first synchronous period [T0, T1xe2x80x2] for sending synchronous signal, and second information period [T1xe2x80x2, T1] for sending information (T0 less than T1xe2x80x2 less than T1). The decrease of information period due to the incorporation of synchronous period will not influence bandwidth because the maximum bandwidth depends on iaj(t)